By considering the commutator, show that the following Hermitian matrices may be simultaneously diagonalized. Find the eigenvectors common to both and verify that under a unitary transformation to this basis, both matrices are diagonalized.
Ω=101000101,Λ=21110−11−12
Since Ω is degenerate and Λ is not, you must be prudent in deciding which matrix dictates the choice of basis.